Optimal estimation of the canonical polyadic decomposition from low-rank tensor trains

Abstract

Tensor factorization has been steadily used to represent high-dimensional data. In particular, the canonical polyadic decomposition (CPD) is very appreciated for its remarkable uniqueness properties. However, computing the high-order CPD is challenging: numerical issues and high needs for storage and processing can make algorithms diverge. Furthermore, the recovery of the CP factors is an ill-posed problem. One way to circumvent this limitation is to exploit the equivalence between the CPD and the Tensor Train Decomposition (TTD). This paper formulates the CPD as a dimension reduction using a TTD followed by a global marginally convex optimization problem. This global optimization scheme estimates the CP factors with minimal error. The resulting approach, Dimensionality Reduction, joint Estimation of the Ambiguity Matrices and the CP FACtors (DREAMFAC), relies on a block-coordinate descent that reaches a first-order stationary point when estimating the CP factors. DREAMFAC is also shown to be an optimal estimator that reaches the corresponding constrained Cramér–Rao bound. It therefore appears as a state-of-the-art solution to estimate the best rank-$K$ CPD of a tensor (when it exists). Its performance is illustrated on the problem of parameter estimation in a dual-polarized MIMO system. Numerical experiments show the excellent practical performance of DREAMFAC, even with very low SNR.